Extracted Text

The following text was automatically extracted from the image on this page using optical character recognition software:

Use of the two former functions leads to summations similar to that 1n (3)which are Invalid. However, use of the phonon free energy per mode■tt u)/2 + k T ln[ 1 - exp(- 3R <u)] In (1) Instead of the B-E function leads toa result that Is well behaved at all temperatures. The purpose of_this work1s to derive the expression for the average free energy per mode f for acrystal having large phonon Unewldths and to test the properties of thethermodynamic functions derivable from f.The procedure 1s to Insert f in (1) and to assume m is complex. The lineIntegration 1s accomplished by contour integration over the w-plane. Thecounterclockwise contour for the upper half plane consists of a line e abovethe real axis (with small semicircle at the origin) plus a large pos1 ’'vesemicircle. Poles occur at w = + 2mri (kT/h); n=l ,2.....but the residuesare zero for all n. The clockwise cortou" for the lower half plane consistsof a line e below the real axis (with small semicircle at origin) plus thelarge negative semicircle. Poles occur at <u = - 2mM (kT/h); n = 1,2,...,but, again, all of the residues are zero. The only residues of Importanceare at the poles id = 13+ 1T (upper half plane) and oj» 13- 1T (lower halfplane). Results for the two half planes are averaged as discussed by Morseand Feshback [2], a procedure which leads to,7 ■ h io/2 + (kT/2) In [1 - 2 exp(- 7) cos y + exp(- 2 7)] , (4)where 7 ■ h ui/kT and y ■ h r/kT. This same procedure was used 1n derivingn 1 n (3) from (1).3. Thermodynamic ApplicationsThe specific heat per mode 1s obtained by the formula c ■ - T 327/3T2.However, the exact form of c depends on the assumptions made as to thetemperature dependence of the anharmonlc shift A(qj) and half width T(qj).Maradudln and Fein [3] derived expressions for A and r and found approxima-tions suitable for the very low temperature range and for the high tempera-ture range with T > Debye G. At low temperatures, A and r are nearly Indepen-dent of T while at high temperatures they vary linearly with T. They alsocalculated formulas for A and T for high temperatures based on an anharmonlcpotential function (Morse potential) which, 1n turn, was based on the heatof sublimation of lead. Using their data, we thus have a basis for testingthe possible validity of (4), at least for the case of crystalline lead.Thus, for the high temperature approximation, we may write,A ■ d T; 7 ■ h wQ/kT + h d/k -1 xQ + 6 ;I’ ■ *j T; y ’ h g/k , (5)where w Is the harmonic (or quasiharmonic) elgentrequency. Using (5) 1n (4),we obtain the mode specific heat,cy ■ k xQ2 o^("(e2^ + 1) cos y ■ 2ex]/(e2x - 2e^cos y + 1)^ (6)Since r Is small In the high temperature approximation, we may replace cos yby (1 - Y7/2). The shift A 1s also small and we may replace exp(x) byexp(x )(1 + <S + 62/2). Thus, (6) can be rewritten 1n the form,cv/k ■ x2 ex/(ex - l)2 - A x2 ex(ex + l)/(ex - 1)^■i (l/2)(i'r - yJ) xL cx(e2x + 4 ex + l)/(eX - 1)^ , (7)1n which x now denotes x0 - h <o0/kT. The leading term 1n (7) 1s the ordi-nary harmonic specific. When Integrated over the spectrum, this leads iothe Dulong-Petlt value of 3R per mole for T > Debye 0, Upon expanding